Theorem confun2 44435

Description: Confun simplified to two propositions. (Contributed by Jarvin Udandy, 6-Sep-2020.)

Hypotheses

Ref	Expression
confun2.1	⊢ (𝜓 → 𝜑)
confun2.2	⊢ (𝜓 → ¬ (𝜓 → (𝜓 ∧ ¬ 𝜓)))
confun2.3	⊢ ((𝜓 → 𝜑) → ((𝜓 → 𝜑) → 𝜑))

Assertion

Ref	Expression
confun2	⊢ (𝜓 → (¬ (𝜓 → (𝜓 ∧ ¬ 𝜓)) ↔ (𝜓 → 𝜑)))

Proof of Theorem confun2

Step	Hyp	Ref	Expression
1		confun2.1	. 2 ⊢ (𝜓 → 𝜑)
2		confun2.2	. 2 ⊢ (𝜓 → ¬ (𝜓 → (𝜓 ∧ ¬ 𝜓)))
3		confun2.3	. 2 ⊢ ((𝜓 → 𝜑) → ((𝜓 → 𝜑) → 𝜑))
4	1, 1, 2, 3	confun 44434	1 ⊢ (𝜓 → (¬ (𝜓 → (𝜓 ∧ ¬ 𝜓)) ↔ (𝜓 → 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206

This theorem is referenced by: (None)